Problem 36 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focal diameter 8 and focus on the negative \(y\) -axis

Step-by-Step Solution

Answer

The equation of the parabola is \(x^2 = 8y\).

Step 1: Identify Vertex and Axis of Symmetry

The vertex of the parabola is located at the origin \((0, 0)\), and since the focus is on the negative \(y\)-axis, the axis of symmetry is vertical. This means that the parabola opens downward.

Step 2: Use the Standard Parabola Equation

Because the parabola opens vertically and downward, its equation in vertex form is given by \(x^2 = -4py\). Here, \(p\) is the distance from the vertex to the focus.

Step 3: Determine the Value of \(p\) from the Focal Diameter

The focal diameter of the parabola (also known as the latus rectum) is 8. The focal diameter is given by \(4|p| = 8\), which can be solved as \(|p| = 2\). Since the parabola opens downward, \(p = -2\).

Step 4: Write the Specific Equation of the Parabola

Substitute \(p = -2\) into the parabola's standard equation. This gives us \(x^2 = -4(-2)y\), simplifying to \(x^2 = 8y\).

Key Concepts

Vertex of a Parabola

When exploring the geometry of a parabola, one of the primary elements is its vertex. The vertex of a parabola represents its 'tip' or the point that is most curved. In the case of a vertical parabola, this is either the lowest or highest point of the parabola. For the parabola discussed in the exercise, the vertex is located at the origin, (0,0).
This point serves as an anchor from which the parabola stretches or curves outward.
Understanding the vertex is crucial because it determines how the parabola is positioned on the coordinate plane:

• The vertex's position influences the symmetry and shape of the parabola.
• The vertex form of a parabola's equation typically reads \(y = a(x-h)^2 + k\), where \((h, k)\) denotes the vertex.
• In our exercise, the equation adapts to \(x^2 = 8y\) because the vertex remains at the origin.

Thus, knowing the vertex allows you to predict how the parabola will behave and where it will lie.

Axis of Symmetry

The axis of symmetry is a vital concept when dealing with parabolas as it defines a line that symmetrically divides the parabola in half.
For a parabola with a vertical orientation like in this exercise, the axis of symmetry will always run vertically, following the line where the parabola 'folds' onto itself.
In this specific example, due to the parabola having its vertex at the origin and opening downward, the axis of symmetry is the vertical line \(x = 0\).
Important points about the axis of symmetry include:

• It helps in visually understanding the parabola's orientation and ensuring that any geometric transformation is appropriately applied.
• For vertical parabolas, it will always be a line parallel to the \(y\)-axis.
• It passes directly through the vertex, thus each point on one side of the axis mirrors onto the other.

This symmetry is key to understanding the balanced and uniform spreading of the curve from the vertex.

Focal Diameter

Focal diameter, also referred to as the latus rectum, provides crucial information about the size and shape of the parabola.
It is defined as the length of the chord passing through the focus, parallel to the directrix of the parabola.
For the parabola in our exercise, the focal diameter is given as 8. It is linked to the parameter \(p\) in the equation \(x^2 = -4py\), where \(4|p|=8\).
This relation allows us to determine \(p\) by solving \(|p| = 2\).
Important features of the focal diameter include:

• It is a fixed distance that is directly tied to the openness or width of the parabola.
• A larger focal diameter suggests that the parabola will be wider or more open.
• For our downward-opening parabola, with a focal diameter of 8, \(p\) is −2, indicating how far the focus is from the vertex along the axis of symmetry.

By understanding this concept, you can infer much about the parabola's dimensions and structure.

Downward-Opening Parabola

Parabolas can open in four main directions depending on their orientation and the sign in their equations.
In this exercise, we focus on a downward-opening parabola, where the parabola opens towards the negative \(y\)-axis.
A few characteristics of downward-opening parabolas include:

• The equation associated is typically of the form \(x^2 = -4py\) which indicates a vertical parabola opening downwards.
• The vertex, positioned at the maximum height of the parabola, results in all points 'drooping' downwards from this origin.
• Its axis of symmetry remains vertical, and each side of the parabola mirrors the other perfectly across this line.

In our specific equation, \(x^2 = 8y\), not only does the sign inform about the downward orientation but also indicates the vertex's role as the parabola reaches most positive on the \(y\)-axis at the origin and then tickets downward.